Plasticity Induced by Shock Waves inNonequilibrium Molecular-Dynamics

SimulationsBrad Lee Holian* and Peter S. Lomdahl

Nonequilibrium molecular-dynamics simulations of shock waves in three-dimensional10-million atom face-centered cubic crystals with cross-sectional dimensions of 100 by100 unit cells show that the system slips along all of the available {111} slip planes, indifferent places along the nonplanar shock front. Comparison of these simulations withearlier ones on a smaller scale not only eliminates the possibility that the observedslippage is an artifact of transverse periodic boundary conditions, but also reveals therichness of the nanostructure left behind. By introducing a piston face that is no longerperfectly flat, mimicking a line or surface inhomogeneity in the unshocked material, it isshown that for weaker shock waves (below the perfect-crystal yield strength), stackingfaults can be nucleated by preexisting extended defects.

Almost 20 years ago, nonequilibriummolecular dynamics (NEMD) simulations,where Newton’s equations of motion aresolved on the computer for thousands ofstrongly interacting atoms, were first usedto study true shock waves at the atomisticlevel (1–4). True shock waves exhibitsteady profiles (density, velocity, stress, andenergy), which accompany dissipative, irre-versible flow of atoms in the directionstransverse to the planar wave propagation.NEMD simulations demonstrated that themethod is particularly ideal for dense-fluidshock waves, where thicknesses are only afew molecular spacings, rise times are only afew collision times, and viscous flow in theshock front leads to steady waves (3). Sub-sequently, Hoover showed that a continuumconstitutive model, using the Navier-Stokesequations of hydrodynamics, could explainthe observed NEMD profiles (5). FurtherNEMD simulations for even stronger fluidshock waves, where the shock thickness cor-responds to only two or three molecularspacings, showed that the Navier-Stokesequations are nevertheless still valid (4).

Whereas viscous flow in dense-fluidshock waves is highly localized, shockpropagation in solids is inherently morecomplex, because solids introduce a newlength scale other than the lattice spacing,namely, the size of defects that governplastic flow. In general, plastic deforma-tion in solids results from the creation andmotion of dislocations (a dislocation is alattice mismatch along a slip plane and ischaracterized by long-range stress andstrain fields). In the early 1980s, shock-

wave structure in solids was elucidated atLos Alamos National Laboratory inNEMD simulations by Holian, Straub, andSwanson (1, 2, 6, 7). These calculationsshowed that planar shock waves in singlecrystals became steady waves throughtransverse displacements of atoms, not byviscous flow as in fluid shock waves (3–5),but rather by plastic flow, that is, concert-ed slippage of atoms over each other. Inthe Lennard-Jones (LJ) pair-potential sol-id, represented by the face-centered cubic(fcc) lattice, a shock wave traveling in the^100& direction results in slippage alongone of the four available {111} planes, byemission at the shock front of a Shockleypartial dislocation, which leaves behind astacking fault (the usual ABCABC . . .stacking of triangular-lattice close-packedplanes becomes ABABCA . . . ).

In order to minimize edge effects andthereby model an infinite plate of materi-al, NEMD shock-wave simulations havetraditionally used periodic boundary con-ditions transverse to the shock-wave prop-agation direction. NEMD calculationsdone as recently as a decade ago (6) wereseverely limited in the length of run inthe direction of shock-wave propagationneeded to achieve a steady wave and inthe transverse cross-sectional area. A typ-ical simulation might be on the order of100 lattice planes in length along theshock propagation direction, and ap-proaching 100 atoms in each cross-sec-tional plane, or 10,000 atoms total inthree dimensions (3D). Computationaltimes are then limited by the sound-tra-versal time in the shock-propagation di-rection, on the order of 25 vibrationalperiods (;5 ps). Shock thicknesses arethen restricted to be less than ;10 nm,with rise times of the order of a picosec-

ond. Thus, weak shock waves, whosethicknesses are measured in fractions ofmicrometers with presumably similar sizesof cross-sectional structures, are well be-yond the reach of atomistic simulations.

In spite of these computational limita-tions in time and space, it is remarkablethat solid shock waves—if they are not tooweak—are nevertheless amenable toNEMD simulations. It is especially note-worthy that the rich details of shock-wavestructure are hardly visible in the set ofshock velocities us obtained in simulationsof various strengths or piston velocities up(Fig. 1). This so-called Hugoniot relationis a global statement of mass, momentum,and energy conservation linking a giveninitial equilibrium state to all possible fi-nal equilibrium states for steady planarshock waves, depending on their strength.It says nothing at all about the structure ofthe shock front, nor anything about dissi-pative structural rearrangement mecha-nisms that could lead to a steady shockwave.

For more localized detail in the shockfront region, we can easily obtain shock-wave profiles from NEMD simulations,such as pressure-volume (PV) tensor com-ponents, normal and transverse to thepropagation direction, and the shear stresst (one-half the difference between thenormal and transverse components of P).Shock-wave profiles from earlier simula-tions (6) in perfect crystals showed thatonce a steady state has been achieved, tbuilds up to a maximum value tmax at the

Theoretical Division, Los Alamos National Laboratory,Los Alamos, NM 87545, USA.

*To whom correspondence should be addressed. E-mail:blh@lanl.gov

Fig. 1. Shock velocity (us/c0) versus piston veloc-ity (up/c0) for atoms in 3D fcc solid (^100& propa-gation direction) interacting via f(r), the intermedi-ate-range LJ spline pair potential (13, 14). Initialdensity is r0 5 =2m/r0

3, where m is atomic mass,r0 is nearest neighbor distance (a0 5 =2r0 is initialfcc lattice constant), «0 is bond energy, and c0 is1D longitudinal sound speed, such that mc0

2 572«0 5 r0

2f0(r0). Initial temperature is kT/«0 50.001. L is cross-sectional periodic boundarylength; the open circles are previous simulationsfor L/a0 5 6 unit cells (6); solid circles are thepresent simulations for L/a0 5 100, which havebeen fitted by straight line us 5 c 1 sup, where c 51.01c0 and s 5 1.86. Note that melting occursnear up/c0 ' 1 (6, 19).

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center of the shock front and is thenrelieved by transverse plastic flow. How-ever, due to the limitation of small com-putational cross sections, these early pro-files were very noisy. Our present simula-tions are for much larger systems andtherefore exhibit much smoother profiles(Fig. 2). For shock strengths at and abovea threshold limit, the ratio of the Hugo-niot jump stress P to tmax appears always tobe P/tmax ; 10 in 3D (6) (although ourpresent large-scale simulations show thatthis ratio increases slowly with shockstrength). At the threshold of shock-in-duced plasticity in perfect crystals, as ob-served in NEMD simulations, the criticalvalue of P ' G, where G is the shearmodulus, so that tmax ' G/10, roughly theideal-crystal yield strength. Provided thatwe could unambiguously rule out the ef-fect of transverse periodic boundary con-ditions, we speculated (6, 7) that preex-isting defects such as vacancies, disloca-tions, or grain boundaries might consider-ably lower this ideal crystal threshold.

Pictures of atomic positions from ourearly NEMD simulations for shocks in per-fect crystals revealed slippage beginning atthe shock front, in the form of what ap-peared to be bands of stacking faults,though the spacing of the bands was clearlydictated by the transverse periodic bound-aries. In fact, only a single slip system wasrandomly selected by thermal fluctuations,so that in principle, the results were consis-tent with the formation of a single stackingfault. For a stronger shock, two parallelstacking faults were seen (6). Long after ourearliest NEMD results, experimental verifi-cation was obtained by Russian investiga-tors, using dynamic shock-wave x-ray mea-surements of shifts in fcc-crystal diffractionpeaks (8). [At the same time, we were doingdynamic shock-wave simulations at LosAlamos, Mogilevsky in the U.S.S.R. (9)

was doing quasi-static relaxation simula-tions in uniaxially stressed fcc crystals andobserving dislocation-emission events simi-lar to those we saw; because of poor con-tacts between U.S. and Soviet scientistsduring the height of the Cold War, bothsides were unaware of the others’ work forseveral years.]

In the early 1990s, at Los Alamos (7), wesimulated shock waves in 2D solids, wherethe cross-sectional area could be taken to benearly an order of magnitude larger thanprevious 3D simulations. We observed thattwo of the three available triangular-latticeslip systems were activated, and the shockfront became irregular, rather than perfectlyplanar as in the early purely elastic phase ofpropagation (7). It remained to be seen,however, whether these 2D observationswould hold for large 3D systems, where thecomplexity of dislocation emission is con-siderably enhanced (10).

Here, we will explore the question ofNEMD system size in evaluating observa-tions of shock-wave–induced plasticity. Re-cent advances in computer technology, pri-marily massively parallel machines and themolecular-dynamics code we have devel-oped (11), enabled us to simulate the ato-mistic behavior of 3D solid materials ex-posed to moderately strong shock waves formuch larger systems (107 to 108 atoms)than were possible just a decade ago.

In NEMD simulations, there are threeprincipal ways to generate a shock wave: (i)As in laboratory planar shock-wave exper-iments, one can computationally hurl a fly-er plate toward a stationary target at a ve-locity of 2up. This is equivalent to slammingthe two plates against each other at 6up; inthis symmetric-impact case, a pair of shockwaves moves out from the interface at 6us.(ii) The symmetric impact can be generatedby inhomogeneously shrinking the longitu-dinal periodic length. This is useful, partic-ularly for fluid shock-wave simulations, foreliminating the effect of free surfaces (4).(iii) Material can be pushed by an infinitelymassive piston moving at velocity up; allparticles coming in contact with the pistonface are specularly reflected by a flat mo-mentum mirror. Equivalently, the pistoncan be at rest, with the unshocked targetmaterial given an initial velocity of –up; theshock wave then moves out from the sta-tionary piston face at velocity us – up. Wehave found that, although there are someminor differences between these threeshock-generation approaches, the first andthird are most suitable for studying bothshock-wave and release-wave phenomenain solids.

Preliminary calculations, which were 15by 15 fcc unit cells in cross-sectional area,demonstrated that instead of a single-slipsystem being triggered by the shock wave,

Fig. 2. Steady shock-wave profile (position/r0) ofpressure-volume tensor (PV/N«0) components:normal to shock plane, and difference betweennormal and transverse (that is, 2 3 shear), at pis-ton velocity up/c0 5 0.5 for square cross-sectionaldimension of L/a0 5 100 unit cells (20,000 atomsin cross-sectional planes).

Fig. 3. Pattern of intersecting stacking faults at piston face (impact plane) induced by collision withmomentum mirror at piston velocity up/c0 5 0.2 for square cross-sectional dimension of L/a0 5 100 unitcells; shock wave has advanced halfway to the rear (;250 planes). Atoms are colored according topotential energy (see color bar at side, energy increasing from bottom).

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slippage occurred along two different {111}-type planes. This behavior is reminiscent ofthe idealized model of C. S. Smith (12),where the shock front in a perfect latticecreates pairs of dislocations that accommo-date the increased density of the shockedmaterial, leaving the crystal orientation vir-tually unchanged, at the same time that theshear stress is relieved. However, the dy-namic results are richer than this staticpicture: the slippages on the two systemscompete with each other, causing the frontto become somewhat unstable, with theleading slip system making the front bulgeout ahead by two or three lattice spacings(as in the 2D case). These slipped regionsare clearly stacking faults, as can be seen bylooking at the oncoming shock front alongthe shock-propagation direction. It appearsthat the interaction potential does not af-fect this shock-induced slippage so much asthe geometry of the fcc crystal structureitself; both LJ pair-potential and embedded-atom-method (EAM) many-body potentialmaterials [such as copper (10, 13, 14)] ex-hibit this behavior. Consistent with theeasier generation of dislocations, however,the EAM system has a lower thresholdshock strength for plastic flow (14).

We tested the hypothesis that preexist-ing point defects could trigger plastic defor-mation by first placing a single vacancy,and then a di-vacancy, in the path of ashock wave, whose strength had been insuf-ficient to initiate plastic flow in the smallcross-section perfect crystal. No plastic de-formation was triggered by either of thesedefects, though there was considerableacoustic disturbance in the front as the

wave passed over them. This is in contrastto shock waves in 2D lattices, where vacan-cies are analogous to line defects in 3D, andtherefore appear to be more effective intriggering plastic flow (15).

We found that the critical transition inshock strength between elastic and plasticbehavior is essentially independent of theinitial temperature T0 [provided that T0 isnot equal to zero, where plasticity appearsto be artificially prohibited, regardless ofshock strength (1)]. We varied the initialtemperature all the way from half the melt-ing temperature TM down to 0.001TM, withno visible effect on the sudden onset ofplasticity with shock strength.

By expanding the cross-sectional area to100 by 100 fcc unit cells in a 10-million-atom simulation, we are now able to answerdefinitively whether the sudden onset ofplasticity observed in NEMD shock waves isreal or an artifact of transverse periodicboundaries. With 4 by 4, 6 by 6, 10 by 10,or 15 by 15 cross sections, we were able toobserve at most two stacking faults in theregion of the transition (up/c0 ' 0.2); now,with 100 by 100, we are able to see that theshock wave propagates about 60 latticeplanes, followed by the relatively suddencreation of a large number of stacking faults.These are distributed randomly on all four{111} slip systems and propagate backthrough the shocked, unslipped material, ata speed that is comparable to the soundspeed in the compressed material, as well asforward with the shock front. As the shockwave propagates further, the front becomes

pronouncedly nonplanar, with the frontbulging by as much as 10 lattice planes.When viewed at an arbitrary {100} plane,the intersections emerge as a randomlyspaced plaid pattern (Fig. 3). This demon-strates conclusively that we are no longerlimited by periodic boundaries, but are see-ing the true nature of plastic flow, at least inthis intermediate regime of shock strength.

When we reduced the shock strengthup/c0 from 0.20 to 0.18, we saw that theplasticity drops sharply to zero. A roughdimensionless measure of plasticity is givenby a0/^,&, where ^,& is the average spacingbetween stacking faults. Above the suddenonset of shock-wave–induced plasticity, a0/^,& follows closely the total volumetricstrain up/us across the shock front (Fig. 4).The dramatic threshold in plasticity for the100 by 100 system, just as in the smallercross-section systems, is clearly due to anonset of ideal crystal yielding, rather than toperiodic boundaries—that is, system size.We confirm this quantitatively by notingthat, at the critical strength where shock-induced plasticity commences, the maxi-mum nonhydrostatic (shear) pressure timeshalf the volume change across the shock isalmost exactly equal to the potential barrierto partial dislocation emission (the unstablestacking fault energy at constant volume),which is 0.16ε0 for the Lennard-Jonesspline potential.

When the shock wave reaches the freesurface and a rarefaction (relief) wave isproduced, the stacking faults that were pro-duced by shock compression are mostly an-

Fig. 4. Shock wave–induced plasticity a0/^,& ver-sus shock strength up/c0 as function of cross-sectional periodic length L/a0 (a0 is the initial fcclattice constant). Shock waves have been initiatedin fcc-lattice ^100&-direction by flat momentummirror at piston face, except as denoted by“warp,” where warped momentum mirror withwavelength L and amplitude h 5 r0 is positioned atx(y) 5 h cos(2p y/L). Resolution in measured plas-ticity is given by error bars, which are smaller thansymbol size for L/a0 5 100; total volumetric strainup/us across shock front is shown as heavydashed curve.

Fig. 5. Stacking faults initiated by collision with warped momentum mirror at piston velocity up/c0 5 0.1for square cross-sectional dimension of L/a0 5 15 unit cells; shock wave has already advanced beyondwindow frame at right. Atoms are colored according to potential energy (see color bar at bottom, energyincreasing from left); periodic box has been tilted around axis of shock propagation direction (left-to-right) by about 30 degrees to show stacking faults.

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nihilated, at least for the 15 by 15 system,where only one or two slip systems areactivated. This is consistent with the obser-vation of much smaller dislocation densitiesin recovered shocked materials (16).

We tested the hypothesis that preexist-ing extended defects could trigger plasticdeformation, by replacing the perfectly flatmomentum mirror at the piston face by awarped mirror. That is, we introduced intothe mirror a wave of amplitude h and wave-length L (the transverse periodic dimen-sion), which, with only two arbitrary pa-rameters, had the effect of mimicking anonplanar impact surface, or of subjectingthe shock front to a preexisting extendeddefect, such as a large inclusion, a disloca-tion, a stacking fault, or a grain boundary.We chose first the case L 5 10a0 and h 5 r0for a shock strength of up/c0 5 0.15, whereno plastic flow had previously been ob-served (at least for shocks initiated at a flat,mirror-smooth piston face in perfect crys-tals, even for L 5 100a0). With the warpedmirror, we observed the production ofstacking faults after the shock wave hadtraveled on the order of 50 lattice planes.This suggests that extended line or surfacedefects will probably trigger plastic flowwhen a planar shock wave passes over them.

At an even lower shock strength, name-ly up/c0 5 0.1, and for L 5 15a0, we againobserved the production of stacking faults(Fig. 5), although it was delayed by almosttwice as long a propagation distance fromthe mirror (;100 planes). These warped-mirror results, mimicking interaction withpreexisting extended defects, continue thetrend of shock-induced plasticity propor-tional to volumetric strain that was seenabove the onset perfect-crystal yielding(Fig. 4). The mode of plasticity is the same,

namely, the production of stacking faults byemission of partial dislocations near theshock front; however, the mechanism ofnucleation has changed from heterogeneousat low strengths, requiring the help of pre-existing extended defects, to homogeneousnucleation, or intrinsic plasticity due toperfect-crystal yielding at higher strengths.

A measure of the shock thickness l isprovided by the volumetric strain rate ε̇ 5up/l as a function of the pressure rise Pacross the shock wave (Fig. 6). The shockthickness is measured from the shear-stressprofile of the steady wave by reading off theexponential decay length (back to zerostress), beginning at the peak value; this isequivalent to constructing a maximal-slopetangent line at the middle of the shockfront for the normal-stress profile, and read-ing off the difference between the inter-cepts for the maximum and minimum stressasymptotes (Fig. 2). The strongest shockappears to reach an asymptotic strain rate;therefore, we fitted the weaker shock wavesto a power law of the form ε̇ ; Pm, wherem 5 3.3. For real metals, where the shockstrengths were much lower than what wehave so far been able to achieve withNEMD (17), the experimental range for mis between 3 and 4.

Thus, it appears more and more likelythat these new large-scale NEMD simula-tions, with the inclusion of realistic inho-mogeneities, will be able to bridge the timeand length scales to macroscopic experi-ments and provide essential input to con-tinuum constitutive models. This conclu-sion has far-reaching implications beyondunderstanding shock-induced plasticity; forexample, large-scale NEMD can be appliedto defect-controlled aspects of shock-in-duced chemistry and detonations (18).

Issues remaining to be resolved include,first of all, the development of a more de-tailed structural model of partial disloca-tions, both in their creation under compres-sive shock-wave loading and in their anni-hilation upon relief-wave unloading fromfree surfaces. Both processes will requirefurther atomistic studies, using systems withlarge cross sections. Second, while it is wellknown (14) that the slope of the Hugoniotcurve (Fig. 1) is related to the anharmonic-ity of the repulsive interaction potential,other potential-related properties are farless well understood. For example, we find,from a sparse set of preliminary simulationsfor EAM systems, that there is a qualitativerelationship between the plasticity thresh-old and the unstable stacking fault energy;that is, both appear to be lower than for thecorresponding effective pair potential.There is a need, as we have done for pair-potential solids, to quantify this relation-ship more thoroughly. Third, because both

elastic wave speed and plasticity depend oncrystal orientation, it will be necessary toprobe fcc perfect-crystal shock response un-der ^110& and ^111& propagation directions,in addition to ^100& already studied. Shockbehavior in other crystal lattices and forother kinds of interactions must also beexplored. Finally, it will be necessary toexpand our repertoire of defects for large-scale NEMD shock-wave simulations, espe-cially to include nanocrystalline samples,where effects of both elastic and plasticanisotropy can be probed.

REFERENCES AND NOTES___________________________

1. B. L. Holian and G. K. Straub, Phys. Rev. Lett. 43,1598 (1979).

2. G. K. Straub, B. L. Holian, R. E. Swanson, Bull. Am.Phys. Soc. 25, 549 (1980).

3. V. Y. Klimenko and A. N. Dremin, in Detonatsiya,Chernogolovka, O. N. Breusov et al., Eds. (AkademiiNauk, Moscow, 1978), p. 79.

4. B. L. Holian, W. G. Hoover, B. Moran, G. K. Straub,Phys. Rev. A 22, 2498 (1980).

5. W. G. Hoover, Phys. Rev. Lett. 42, 1531 (1979).6. B. L. Holian, Phys. Rev. A 37, 2562 (1988).7. iiii, Shock Waves 5, 149 (1995).8. E. B. Zaretsky, G. I. Kanel, P. A. Mogilevsky, V. E.

Fortov, High Temp. Phys. 29, 1002 (1991).9. M. A. Mogilevsky, in Shock Waves and High Strain

Rate Phenomena in Metals, L. E. Murr and M. A.Meyers, Eds. (Plenum, New York, 1981), p. 531.

10. S. J. Zhou, D. M. Beazley, P. S. Lomdahl, B. L.Holian, Phys. Rev. Lett. 78, 479 (1997); S. J. Zhou,D. M. Beazley, P. S. Lomdahl, A. F. Voter, B. L.Holian, in Ninth International Conference on Frac-ture, B. L. Karihaloo et al., Eds. (Pergamon, Sydney,1997), p. 3085.

11. The SPaSM (“Scalable Parallel Short-Range Molec-ular Dynamics”) code is described in: P. S. Lomdahl,P. Tamayo, N. Grønbech-Jensen, D. M. Beazley,Proceedings of Supercomputing 93, IEEE ComputerSociety Press, 520 (1993); D. M. Beazley and P. S.Lomdahl, Parallel Comput. 20 173 (1994); D. M.Beazley and P. S. Lomdahl, Comput. Phys. 11, 230(1997).

12. C. S. Smith, Trans. Metall. Soc. AIME 212, 574(1958); J. W. Taylor, J. Appl. Phys. 36, 3146 (1965);J. J. Gilman, Appl. Mech. Rev. 31, 767 (1968).

13. B. L. Holian and R. Ravelo, Phys. Rev. B 51, 11275(1995).

14. B. L. Holian et al., Phys. Rev. A 43, 2655 (1991).15. R. L. Blumberg Selinger, private communication.16. E. Zaretsky, Acta Metall. Mater. 43, 193 (1995); M. A.

Meyers, Scripta Metall. 12, 21 (1978).17. J. N. Johnson and L. M. Barker, J. Appl. Phys. 40,

4321 (1969); J. R. Asay and L. C. Chhabildas, in (12),p. 417; J. Lipkin and J. R. Asay, J. Appl. Phys. 48,692 (1985); J. W. Swegle and D. E. Grady, ibid. 58,692 (1985); D. Tonks, in Shock Waves in CondensedMatter, S. C. Schmidt and N. C. Holmes, Eds.(North-Holland, Amsterdam, 1987), p. 231.

18. C. T. White, D. H. Robertson, M. L. Elert, D. W.Brenner, in Microscopic Simulations of Complex Hy-drodynamic Phenomena, M. Mareschal and B. L.Holian, Eds. (Plenum, New York, 1992), p. 111;D. W. Brenner, D. H. Robertson, M. L. Elert, C. T.White, Phys. Rev. Lett. 70, 2174 (1993).

19. A. B. Belonoshko, Science 275, 955 (1997).20. It is a distinct pleasure to thank our colleagues J.

Hammerberg, D. Beazley, S. Zhou, and R. Ravelo.We also benefited from discussions with A. Be-lonoshko, E. Zaretsky, P. Makarov, R. Selinger, R.Mikulla, T. Germann, W. Hoover, C. White, and R.Thomson. The work at Los Alamos was encouragedby S. Younger and supported by the U.S. Depart-ment of Energy.

24 February 1998; accepted 12 May 1998

Fig. 6. Strain rate up/l (in units of t0–1) versus

shock pressure P (in units of r0c02) as function of

cross-sectional periodic length L; open circlesare previous simulations for L/a0 5 6 unit cells;solid circles are present simulations for L/a05 100, with power-law fit for weaker shockstrengths shown as straight line (slope 5 3.3). upis piston velocity, l is shock thickness, r0 is initialdensity, c0 is 1D longitudinal sound speed, andt0 5 r0(m/«0)1/2.

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